170 DETERMINATION OF AN ORBIT FROM [BOOK II. 



unless X, Y, have vanished for the second. Then, the true values will be de 

 duced by means of the formulas of the preceding article, so far as the assumption 

 on which these formulas are based, does not differ sensibly from the truth. In 

 order that we may be better able to judge of which, the calculation of the values 

 of X, Y, will be repeated with those corrected values ; if this calculation shows 

 that the equations .X&quot;= 0, F= 0, are, still, not satisfied, at least much smaller 

 values of X, Y, will result therefrom, than from the three former hypotheses, and 

 therefore, the elements of the orbit resulting from them, will be much more exact 

 than those which correspond to the first hypotheses. If we are not satisfied 

 with these, it will be best, omitting that hypothesis which produced the greatest 

 differences, to combine the other two with a fourth, and thus, by the process of 

 the preceding article, to obtain a fifth system of the values of x, y ; in the same 

 manner, if it shall appear worth while, we may proceed to a sixth hypothesis, 

 and so on, until the equations X 0, Y= 0, shall be satisfied as exactly as the 

 logarithmic and trigonometrical tables permit. But it will very rarely be neces 

 sary to proceed beyond the fourth system, unless the first hypotheses were very 

 far from the truth. 



122. 



As the values of the unknown quantities to be assumed in the second and third 

 hypotheses are, to a certain extent, arbitrary, provided, only, they do not differ 

 too much from the first hypothesis ; and, moreover, as care is to be taken that the 

 ratio (a&quot; - a) : (b&quot; - b) does not tend to an equality with ( a) : (b b], it is 

 customary to put =, b&quot;=b. A double advantage is derived from this; for, not 

 only do the formulas for , 77, become a little more simple, but, also, a part of the 

 first calculation will remain the same in the second hypothesis, and another part 

 in the third. 



Nevertheless, there is a case in which other reasons suggest a departure from 

 this custom : for let us suppose X to have the form X x, and Y the form 

 Y -y, and the functions X , Y , to become such, by the nature of the problem, 

 that they are very little affected by small errors in the values of x, y, or that 



A! X \ (dX \ /dT\ /d T \ 

 \dx/ \dy/ \dx/ \dy 



